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On Semi-Suslin Spaces and Dual Metric Spaces
Functional Analysis, Holomorphy and Approximation Theoty, JA. Barroso led.) 0North-HollandF’ublishing Company, I982
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
Manuel V a l d i v i a
I n t h i s p a p e r we s t u d y some p r o p e r t i e s of a c l a s s of t o p o l o g i c a l s p a c e s i n c l u d i n g t h e K-Suslin
s p a c e s and h e n c e f o r t h we o b t a i n
some new r e s u l t s i n t h e t o p o l o g i c a l v e c t o r s p a c e s t h e o r y .
A i s a s t a r s h a p e d m e t r i z a b l e subse-t of a
l a r l y , we p r o v e t h a t i f Hausdorff
E,
topological v e c t o r space
E
bounded s u b s e t of
Particu-
intersects
A
s u c h t h a t e v e r y c l o s e d and
i n a compact s e t , t h e n
A
is
separable.
We u s e t h e r e H a u s d o r f f t o p o l o g i c a l s p a c e s . v e c t o r s p a c e s u s e d h e r e a r e d e f i n e d on t h e f i e l d complex numbers. B(E,F)
and
respectively
3
If
(E,F)
If
W(E,F)
K
of t h e r e a l or
i s a d u a l p a i r we d e n o t e by
t h e weak,
.
The t o p o l o g i c a l
u(E,F),
s t r o n g and Mackey t o p o l o g i e s on
i s t h e t o p o l o g y of a t o p o l o g i c a l s p a c e
ACT]
T
and
E,
a s u b s e t of
T,
we d e n o t e by
t o p o l o g y by
3.
The t o p o l o g i c a l d u a l of a l o c a l l y convex s p a c e
is
E‘
.
X(E’,E)
E”
the s e t
i s t h e t o p o l o g i c a l d u a l of
t h e t o p o l o g y on
compact s u b s e t s of
E.
E’ On
with the
A
E’[B (E‘ , E ) ]
E
induced E
We d e n o t e by
of t h e u n i f o r m c o n v e r g e n c e on t h e p r e -
E”
,
X(E” ,E’
)
i s t h e t o p o l o & T of t h e
u n i f o r m c o n v e r g e n c e on t h e p r e c o m p a c t s u b s e t s of u s u a l , we i d e n t i f y
.
is
A
w i t h a s u b s p a c e of
EN
E‘[B (E’ , E ) ]
.
As
by t h e c a n o n i c a l i n -
j e c t i on. By a “ w e a k l y u-compact
g e n e r a t e d l o c a l l y convex s p a c e ”
G
446
M.
VALDIVIA
we mean a l o c a l l y convex s p a c e
which h a s a s e q u e n c e of weakly
G
compact s u b s e t s whose u n i o n i s t o t a l i n of bounded
of
B
s o that i f
I.
H
z
H
E B,
i s contained i n
{txo
+
(1-t)z
: 0 i t
A t o p o l o g i c a l space
a P o l i s h space s e t s of
2.
E
space
{x,]
and a mapping
x
c
I.
in
B
l] C B .
P of
cp
and a mapping
u
{cp(x) : x
c
cp
from
P
there exists
i n t o t h e c l o s e d sub-
in
P
PROPOSITION 1.
so that
t h e r e i s a P o l i s h space E,
then
from
V
P
(zn)
h a s an adherent p o i n t
i f there is a Polish
i n t o t h e compact s u b s e t s of F ,
and g i v e n a n a r b i t r a r y p o i n t of
z
t h e r e i s a neighbourhood
cp(z),
[7].
cp(U) C V ,
P
converging towards an element
i s K-Suslin
F
A topological
compact s u b s e t s of
P
cp(x).
P] = F
and a n e i g h b o u r h o o d z
i s semi-Suslin i f
n = 1,2,...,
which i s c o n t a i n e d i n
P
[lo]:
E
i s a seqaence i n
zn E cp(xn),
such t h a t
U
i
such t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :
If
and E
P
A topological space
in
I]
SEMI-SUSLIN SPACES
DEFINITION.
in
2
c
is f u n d a m e n t a l
i s starshaped i f t h e r e i s a vector then
: i
{Ai
for some
A.
The f o l l o w i n g d e f i n i t i o n was g i v e n i n
x
H
s e t s i n t h e topological vector space
i f e a c h bounded s e t i n A subset
A family
G.
space
E
i s K-Suslin
and a mapping
cp
from
i f and o n l y i f P
into the
s u c h t h a t t h e two f o l l o w i n g c o n d i t i o n s a r e
satisfied:
2.
x
and
If zn
(x,)
i s a sequence i n
E cp(x,),
n = 1,2,...,
P
c o n v e r g i n g t o w a r d s an e l e m e n t
then
(zn)
h a s an a d h e r e n t p o i n t
447
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
in
which i s c o n t a i n e d i n
E
PROOF.
Let u s suppose t h a t E
p r o p o s i t i o n and x
E
P,
U
of
(Un)
L e t u s t a k e now a p o i n t
z n E cp(xn),
g
zn
t i o n says t h a t cp(x).
E
v e r i f i e s t h e two c o n d i t i o n s of t h e
i s n o t a K-Suslin
a neighbourhood
neighbourhoods
to
cp(x).
xn
in
E
(zn)
P
Un
n = 1,2,
U,
~ ( x ) and a f u n d a m e n t a l s y s t e m of
of x
Then t h e r e i s a p o i n t
space.
z
n=1,2,... and a
The c o n d i t i o n 2 of t h e p r o p o s i -
z
h a s an a d h e r e n t p o i n t
On t h e o t h e r h a n d ,
U,
cp(xn) q! U,
such t h a t
...
~ ( u , )$
such t h a t
6
in
~ ( x )because
E zn
which b e l o n g s
E
U,
n=1,2
,...
T h e r e f o r e we a r r i v e t o a c o n t r a d i c t i o n . E
C o n v e r s e l y , l e t us suppose t h a t a mapping from a P o l i s h s p a c e
u
that
and a n e i g h b o u r h o o d U
of
= E
(cp(x) : x f P]
z
in
P
of
V
such t h a t
i s K-Suslin.
i n t h e compact s u b s e t s o f
P
~ ( z )i n
E,
cp(U) c V .
x
and a s e q u e n c e
which h a s n o t a d h e r e n t p o i n t i n
compact, i f
M
in
cp(x).
,...,zn ,...3 ,
M
t e and t h e r e f o r e t h e r e e x i s t s a p o s i t i v e i n t e g e r
... ] n cp(x)
( ~ ~ ~ , z ~ ~ + ~ E, v.i d. e. n]t l.y , bourhood
X
of
x
positive integer znl
E
B fl cp(x
such t h a t n1
n
B
Let
~ ( x = ) $,
B
) c B fl q ( X ) = $ ,
in
Since
n
no
cp(x)
On t h e o t h e r h a n d ,
which a r e n o t K-Suslin
[lo].
P
cp(x) i s is fini-
such t h a t
hence t h e r e i s a neighWe c a n c h o o s e now a
so that
x
Therefore
P r o p o s i t i o n 1 allow u s t o o b t a i n t h a t e v e r y K-Suslin i s semi-Suslin.
does
zn 6 ~ ( x , ) ,
E,
E X. "1 hence a c o n t r a d i c t i o n .
n
P
b e t h e c l o s u r e of
cp(X) r7 B = $ .
l a r g e r than
"1
= Q.
(x,)
(zn)
n = 1,2,...
no+P'
in
cp
Let u s suppose t h a t
which converges towards
rzno,zno+l,...,z
such
t h e r e i s a neighbourhood
Then t h e r e i s a s e q u e n c e
{z1,z2
be
E
z
and g i v e n a n a r b i t r a r y p o i n t
n o t v e r i f y c o n d i t i o n 2.
i s the s e t
cp
Let
q.e.d. space
t h e r e a r e semi-Suslin spaces
448
M.
LEMMA 1.
Let
F
VALDIVIA
be a s e m i - S u s l i n t o p o l o g i c a l s p a c e .
F
m e t r i z a b l e c l o s e d s u b s p a c e of
then
E
If
is a
E
i s K-Suslin.
PROOF.
S i n c e e v e r y c l o s e d s u b s p a c e of a s e m i - S u s l i n s p a c e i s semi-
Suslin,
[lo],
w e t a k e a mapping
L e t us t a k e i n
compact.
n = 1,2,.
..
cp(x)
Let
(U,)
t h e sequence
P,
i n the
(zn)
~ ( x )i s
then
(xn)
xn = x ,
siich t h a t
i s a n a r b i t r a r y sequence i n
cp (x)
,
zn F cp(xn),
has an a d h e r e n t p o i n t i n
i s c o u n t a b l y compact.
results that
Let
P
and t h e r e f o r e
rp(x)
LEMMA 2 .
(zn)
If
n = 1,2,..., Hence
i s an a r b i t r a r y point i n
x
P
T a k i n g a c c o u n t of P r o p o s i t i o n 1, i t s u f f i c e s t o
tion are verified. show t h a t i f
from a P o l i s h space
s o t h a t t h e c o n d i t i o n s 1 and 2 of t h e d e f i n i -
E
c l o s e d s u b s e t s of
rp
Since
i s metrizable i t
E
i s compact.
q.e.d.
b e a s u b s e t of a t o p o l o g i c a l v e c t o r s p a c e
A
rp(x).
E
b e a s e q u e n c e o f c l o s e d c i r c l e d s u b s e t s of
E[Z].
which v e -
r i f i e s t h e following conditions:
1.
If
z
an a r b i t r a r y p o i n t
of
a r e given t h e r e i s a positive i n t e g e r 2.
If
(mp)
and a p o s i t i v e i n t e g e r
A
n
P
such t h a t
z
E
p
n U P P'
i s a n a r b i t r a r y s e q u e n c e of p o s i t i v e i n t e g e r s t h e
set
[n i s non-void
Then PROOF.
Let
Empup
= i,2, ...I]
n
A
and c o u n t a b l y compact. A[3] N
i s a semi-Suslin space.
b e t h e s e t of p o s i t i v e i n t e g e r numbers w i t h t h e
d i s c r e t e topology.
By
NN
of c o u n t a b l e many c o p i e s of
u s c o n s i d e r t h e mapping A[J]
: P
rp
we r e p r e s e n t t h e t o p o l o g i c a l p r o d u c t N.
Then
from
NN
NN
i s a P o l i s h space.
i n t o t h e c l o s e d s u b s e t s of
such t h a t i f
x = (x1,x2
Let
,...,xP' ...) E
NN
44 9
ON SEMI-SUSLIN S P A C E S AND DUAL METRIC SPACES
then
CP(~)rn Expup :
...I] n
P = i,2,
A.
Using condition 1 of this lemma it follows that A = [pp(x)
(x'"))
converges towards
If
cp(x(")). teger
n(p)
p
2
x
in
NN
NN.
such that the sequence zn
Let us take a point
such that
x(~) = x
P (x("))
for every P' towards x in
n 2 n(~),
NN.
because
Then, if
n(p), zn E cp(x ( n ) ) c xPuP
{.An)
...I
: n = 1,2,
then
zn
E
[n CYpup
: P = i,2,
and since this set is countably compact, point
in
zo
A[3]
by (1).
has an adherent
...!I
: P = i,2,
n
A = rp(x)
q.e.d.
THEOREM 1.
Let
be a starshaped metrizable subset of a topolo-
A
E[3].
gical vector space intersects
E
(zn)
A
Therefore conditions 1 and 2 of the definition are sa-
tisfied.
PROOF.
...11 n
which belongs to
3? Expup
E[3]
(1)
is the maximum of the finite set of natural numbers
If yp
of
in
is a positive integer there exists a positive in-
of the convergence of n
,...,xP( " ) ,...) E
= (xin),x$n)
x(")
Let
N
: x E N ).
A
If every closed and bounded subset of
in a compact set,
A[3]
is a K-Suslin space.
Obviously, it sufficos - t o prove the theorem when the origin belongs to
point of
A
A
and every segment which joints an arbitrary
with the origin lies in
there is a sequence
(U,)
A.
Since
A
is metrizable
of closed circled neighbourhoods of the
M. VALDIVIA
450
origin i n
such t h a t
E[3]
(un n is
R
f u n d a m e n t a l s y s t e m of
A : n=1,2,
...I
n e i g h b o u r h o o d s of t h e o r i g i n i n
O b v i o u s l y c o n d i t i o n 1 o f Lemma 2 i s v e r i f i e d on
(Un).
If
A[3]. (mp)
i s a n a r b i t r a r y s e q u e n c e of p o s i t i v e i n t e g e r s t h e s e t
i s non-void,
because t h e o r i g i n has i n i t .
To p r o v e t h i s l e t
t h i s s e t i s compact.
bourhood of t h e o r i g i n i n
q
Let u s s e e now t h a t
V
b e an a r b i t r a r y neigh-
Then t h e r e i s a p o s i t i v e i n t e g e r
E[3].
such t h a t
u 9 n ~ c v An. If
z
E
n
(mqUq)
A,
n
E Uq
then
A
and i t f o l l o w s t h a t
¶
[n
Empup
: p=1,2,
and t h e r e f o r e ,
...I] n
A C
( m9 u9 ) n
A C m ( U nn) q 9
t h e s e t ( 2 ) i s bounded i n
E[3].
i t i s evident t h a t t h i s s e t i s closed i n i s compact. verified.
I t allows u s t o conclude t h a t
PROOF.
v
9
O n t h e o t h e r hand,
and, t h e r e f o r e ,
it
intersects
A[%]
i s a semi-Suslin
i s K-Suslin.
A[3]
b e a s t a r s h a p e d and m e t r i z a b l e s u b s e t o f a t o -
A
Let
pological vector space E[J]
9
We h a v e t h u s p r o v e d t h a t c o n d i t i o n 2 of Lemma 2 i s
s p a c e , and by Lemma 1, THEOREM 2 .
A[3]
c rn ( V n A ) c m
E[3].
If
e a c h bounded and c l o s e d s u b s e t of
i n a compact s e t ,
A
By Theorem 1,
A[3]
i s K-Suslin
A[3]
i s separabla.
and t h e r e f o r e L i n d e l 8 f , [ 7 ] .
Since every metrizable Lindelbf space i s separable, complet e , COROLLABY 1 . 2 .
t h e proof
is
q.e.d. Let
g i c a l v e c t o r space
A
E[3].
b e a m e t r i z a b l e convex s u b s e t o f a t o p o l o I f e v e r y bounded and c l o s e d s u b s e t of
451
ON SEMI-SUSLIN SPACES AND DUAL METRIb SPACES
E[3]
intersects
NOTE 1.
A
in a compact set,
A. Grothendieck asks in
A[s]
is separable.
[4] if every FrBchet-Monte1 space
J. Dieudonne gave an affirmative answer to this
is separable.
.
question in [3]
C. Bessaga and S. Rolewicz proved in [ 2 ]
that
every metrizable Monte1 topological vector space is separable. This result can be obtained from our Corollary 1.2 taking THEOREM
3
3.
Let
A = E.
be a metrizable topological vector space.
E
be a topological vector t o p o l o g on
E
Let
coarser than the ori-
ginal topology such that the following conditions are satisfied: 1.
There is a fundamental system of neighbourhoods of the E
origin of 2.
which are closed in
E[3].
Every bounded subset of E is relatively countably compact in
EC3l. E[3]
Then PROOF.
is a semi-Suslin topological space.
(Un) be a fundamental sequence of circled neighbour-
Let
hoods of the origin in A = E
E,
which are closed in
that lemma is satisfied.
On the other hand, let
quence of positive integer numbers.
n compact.
E[3]
(mp)
be a se-
The set
Empup : p=i,2, ...I
and closed in
E[3]
and, therefore, 3-countably
Consequently, condition 2 is satisfied.
It follows that
is a semi-Suslin space.
THEOREM 4. 3
E
Let us take
I t follows straightforward that condition 1 of
in Lemma 2 .
is bounded in
E[3].
Let
E
be a metrizable topological vector space.
be a topological vector topology on
E
Let
coarser than the original
topology such that the following conditions are satisfied:
1.
There exists a fundamental system of neighbourhoods of the
origin in
E
which are closed in
E[3].
452
M.
E v e r y bounded s e t i n
2.
Then
E[3]
i s a K-Suslin
i s r e l a t i v e l y compact i n
E[3].
topological space.
of Lemma 2 i s s u c h t h a t
b u i l t i n t h e proof
x E N
f o r every
x = (x1,x2,
E
EC31.
9.e.d.
THEOREM
5.
If
.
rp(x)
i s compact i n
It follows s t r a i g h t f o r w a r d , s i n c e i f
the s e t
and c l o s e d i n
E
and t h e r e f o r e compact i n
E[3]
i s a Fr6chet space,
E'[b(E'
i f and o n l y i f PROOF.
N
...,xn, ...)
i s bounded i n
A
E
By P r o p o s i t i o n 1, i t s u f f i c e s t o p r o v e t h a t t h e mapping
PROOF.
E[3]
VALDIVIA
,EN)]
E"[X(E",E')]
i s K-Suslin
i s barrelled.
L e t us s u p p o s e f i r s t t h a t
i s K-Suslin.
E"[X(E" , E ' ) ]
Let
b e a n a b s o l u t e l y corivex c l o s e d and bounded s u b s e t o f E " [ u ( E " , E ' ) ] .
G r o t h e n d i e c k proved t h a t e v e r y c o u n t a b l y s u b s e t of tinuous i n E"[u(E"
Since
,E')].
E"[a (E" ,E' )]
[4],
[ 71
E'[w (E' ,E"
hence
.
)]
and t h e r e f o r e Hence
E"[p(E" , E ' ) ] ,
S i n c e each sequence i n
i t follows t h a t
).(E",E')
A
i s a Lindel8f
i s u(E" ,E' )-compact and c o n s e -
A
E'[u(E'
,E")]
is barrelled.
A
and
u(E",E') A
If
E ' [ p (E' , E ) ]
E
is
[S]
,
c o i n c i d e s on t h i s sequence,
i s r e l a t i v e l y compact i n
By u s i n g Theorem 4 w e o b t a i n t h e c o n c l u s i o n . Let
A
i s r e l a t i v e l y u(EN, E ' ) - c o m p a c t .
i s equicontinuous i n
hence i t i s e a s y t o o b t a i n t h a t
COROLLARY 1.5.
A[a (E" , E ) ]
of t h e o r i g i n which a r e h ( E " , E ' ) - c l o s e d .
a bounded s e t i n
,E')].
it follows t h a t
i s a F r 6 c h e t s p a c e which h a s a f u n d a m e n t a l s y s t e m of
neighbourhoods
E"[X(E"
i s c o u n t a b l y compact i n
i s barrelled.
L e t u s s u p p o s e now t h a t
E"[p(E",E')]
A
i s K-Suslin
E"[h(E" , E ' ) ]
i s K-Suslin
t o p o l o g i c a l space quently
,
E'[@(E' ,E)]
i s equicon-
A
be a Fr6chet space.
If
E
q.e.d.
i s distinguished
453
ON SEMI-SUSLIN SPACES AND DUAL &QCTRICSPACES
then
I n [4]
NOTE 2.
G
(EN ,E' )]
E"[
i s K-Suslin. G r o t h e n d i e c k g i v e s a n e x a m p l e s of a F r 6 c h e t s p a c e
A.
G'[p (G' ,G" )]
such t h a t
i s n o t K-Suslin
i s not barrelled.
H
example o f a F r 6 c h e t s p a c e $(HI ,H)
and
f
,*). H
w(H'
g ' [ X ( ~, H ' ')]
THEOREM 6.
E
Let
PROOF. that
If
such t h a t
Komura g i v e s i n
E"[X(E",E')]
i s K-Suslin
i s Lindelbf. A
and t h e r e f o r e
E"[X(E"
Since
,E')]
is
E"[
i s an
A
(E" ,E' ) ]
(E" ,E' ) - c o u n t a b l y
,
compact
Hence E'[p (E' ,E" )]
q.e.d.
DUAL METRIC SPACES A l i n e a r topological
if
A
is
E"[X(E",E')]
i s L i n d e l b f and
i s X (EN ,E' ) - c o m p a c t .
is barrellled.
11.
5
i s b a r r e l l e d i t f o l l o w s f r o m Theorem
E'[p(E',E")]
[ 41 i t f o l l o w s t h a t
i s Lindelbf
is barrelled.
E'[p(E',E")]
Conversely, i f
)]
an
i s barrelled
H'[M(H',H")]
a b s o l u t e l y convex c l o s e d and boJnded s u b s e t o f
A[A (E" ,E'
[6]
i s K-Suslin.
be a Fr6chet space.
E"[X(E",E')]
Lindelbf.
Y.
i s a n example of a n o n - d i s t i n g u i s h e d
space such t h a t
i f and o n l y i f
5.
b e c a u s e o f Theorem
T h e r e f o r e G"[X (G" ,G')
l o c a l l y convex s p a c e
E
i s dual metric
i t h a s a c o u n t a b l e f u n d a m e n t a l s y s t e m of bounded s e t s and i n E'
each
3 (E' ,E)-bounded s e q u e n c e i s e q u i c o n t i n u o u s [ 91 , p . 11. A l i n e a r t o p o l o g i c a l l o c a l l y convex s p a c e
E
i s (DF)
h a s a c o u n t a b l e f u n d a m e n t a l s y s t e m of bounded s e t s a n d i n $(E',E)-bounded
E'
i f it each
s e t which i s c o u n t a b l e u n i o n of e q u i c o n t i n u o u s s e t s
i s i t s e l f equicontinuous [4]. Obviously,
e v e r y (DF)-space i s d u a l m e t r i c .
The f o l l o w i n g
two t h e o r e m s g i v e some c l a s s e s o f d u a l m e t r i c s p a c e s which a r e n o t (DF) *
454
M.
THEOREM
7.
pology
3
If
compatible with the dual p a i r
E',
(E,E')
such t h a t
i s a d u a l m e t r i c s p a c e which i s n o t (DF).
E'[3]
Let
PROOF.
Since
i c
I]
b e a maximal o r t h o n o r m a l s y s t e m i n
i s not separable t h e r e i s a p a r t i t i o n of
E
...,I n ,...
11,12,
n = 1,2,... : i
:
(xi
many s u b s e t s
(xi
i s a non-separable H i l b e r t space t h e r e i s a t o -
E
on
VALDIVIA
Let
6 In]
In
i n countable
i s not countable,
b e t h e c l o s e d a b s o l u t e l y convex h u l l of
An
in
such t h a t
I
E.
E.
We d e n o t e by
@
a l l t h e s u b s e t s of
E
of
t h e form m
with
a b s o l u t e l y convex bounded and s e p a r a b l e and
A
n i t e l y many n o n - z e r o p o l o g y on
E'[3]
3
If
3
i s t h e to-
i s compatible w i t h t h e d u a l p a i r
i s a dual metric space.
(DF)-space because
3
{A,
: n=1,2,
Moreover,
...I
i t i s not i t s e l f equicontinuous.
(E,E')
i s not a
E'[3]
E'[3]
7 a r e not
Theorem 8 g i v e s Mackey d u a l m e t r i c s p a c e s which a r e
(DF).
F o r t h e n e x t theorem l e t u s t a k e a Banach s p a c e t h e r e i s a n o n - s e p a r a b l e a b s o l u t e l y convex and we-kly set
and
q.e.d.
The d u a l m e t r i c s p a c e s o b t a i n e d u s i n g Theorem Mackey s p a c e s .
8,
i s a s t r o n g l y bounded s e t
w h i c h i s a c o u n t a b l e u n i o n of e q u i c o n t i n u o u s s e t s i n
not
a fi-
o f t h e u n i f o r m c o n v e r g e n c e on t h e e l e m e n t s of
E'
i t i s evident t h a t and
s e q u e n c e of r e a l numbers.
(X,)
X
in
F.
Let
E
t i n u o u s f u n c t i o n s from
b e t h e Banach s p a c e X[o(F,F')]
into
F
compact s u b -
C(X[O(F,F')])
K,
so that
of con-
w i t h t h e u n i f o r m con-
vergence topology. THEOREM 8. such t h a t
In
E"
t h e r e i s a v e c t o r subspace
E'[u(E',L)]
L
containing
i s a non-(DF) d u a l m e t r i c s p a c e .
E
455
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
PROOF.
Let
M
b e t h e l i n e a r h u l l of
is
M
t h a t t h e o r i g i n of
in
Gb
X
in
X[a(F,F')].
.
m
n
Let u s suppose Let
b e a se-
(Un)
of t h e o r i g i n i n
q u e n c e of a b s o l u t e l y convex n e i g h b o u r h o o d s X[u(F,F' )]
F.
Un = { O } For each p o s i t i v e i n t e g e r p n= 1 { p Un : n = 1 , 2 , ...I i n t h e f a m i l y of n e i g h b o u r h o o d s of t h e o r i g i n
such t h a t
( p X)[u(F,F')]
h a s t h e same p r o p e r t y .
Given t h e p a i r
positive integers there exists a f i n i t e set
A
in
P"
(p,n) F'
of
such t h a t
Ao n p X c p U n pn
being
t h e p o l a r s e t of
Ao P"
If
P
A
in
Pn
i s t h e l i n e a r h u l l of
(M,P)
i s a d u a l p a i r and
Since
P
u(M,P)
F.
: p , 1 , 2 ,...)
(Apn
coincides with
u(F',F)
h a s a c o u n t a h l e Hamel b a s i s i t f o l l o w s t h a t
i s m e t r i z a b l e and, t h e r e f o r e , s e p a r a b l e .
in
F', in
X.
X[u(F,F')]
But t h i s a c o n t r a d i c t i o n
w i t h the h y p o t h e s i s .
It permits t o a s s e r t t h a t t h e r e i s a point which i s n o t
X[u(F,F')]
s y s t e m of n e i g h b o u r h o o d s of f u n c t i o n from fi(xo) we t a k e
= 1,
X[o(F,F')]
= 0,
fi(x)
a bounded n e t i n
E.
x E X-
(g,ux)
= g(x),
(f,ux)
= 0, Let
H E 51,
let
51 H*
If
in
be a f u n d a m e n t a l fi
[O,l]
il,iz E I (fi
be a continuous such t h a t and vil : i E I, S
t h e r e i s a element
f
in
c v
iz is
)
E"
t o this net.
let
f o r every
I}
and l e t
X
vi.
Consequently,
x 6 X,
x E X-
in
E
i s a d i r e c t s e t and
w h i c h i s a(E" ,E' ) - a d h e r e n t For each
xo
: i
(Vi
i n t o the i n t e r v a l
(I, s)
iz i il.
Let
Gb.
x
ux
b e t h e e l e m e n t of
g E C(X[u(F,F')]).
such t h a t
E'
Obviously,
(f,uXO)=l,
(xo3.
b e the f a m i l y of a l l t h e c o u n t a b l e s e t s i n b e t h e c l o s u r e of
H
in
E"[u(E" ,E')]
.
E. Let
If
456
M. VALDIVIA
L
is a subspace of
E"
containing
Then there is a sequence
,E' ) ]
E"[u(E"
point in
wn The set r) {W,
(g,)
.
which has
E
f
as adherent
Let
x
= cx E
: n=1,2,.
..]
{x,]
is different from
and therefore
such that 1 n
<
Ign(xl)-gn(xo)
1 r ~ .
<
: ign(x)-gn(xo)i
x1 f xo
x1 E X,
there is a point
in
Let us suppose that f E L.
E.
11
9
= 1,2,...
It follows that
(f,ux ) 1
-
and this gives a contradiction.
A
set of H*
3
0
i
= 0
Therefore,
E'[t.i(E',L)]
Let us see now that Let
(f,ux
f
{
L.
is a dual metric space.
be a separable absolutely convex, closed and bounded sub-
aU(L,E')].
A,
hence
A
is u(L,E')-compact
equicontinuous and so,
E'[k(E,L)]
and therefore
such that is p(E',L)-
A
is a dual metric space.
In the sequel we shall prove that (DF)-space.
H € 51
Then there is an element
E'[@(E',L)]
is not a
B y using a result of Amir and Lindestrauss [l] there
D
of
is dense in
B.
is an absolutely convex weakly compact and total subset Let
B
be the closed unit ball of
E.
E.
Let
Bn = B n n D. Bn
is weakly compact and
L
Since
is different of
not contained in
L,
B*
n
L
B*
n
L
u [Bn : n=1,2, ...] that u [Bn : n=1,2, . . . I
On the other hand,
Bn
B*
of
B
in
is a 8 L,E')-bounded
and, consequently,
Since
it follows
continuous.
{Bn : n=1,2,.. ]
E N , the closure
hence
is not g(L,E')-cornpact equicontinuous.
u
BY
n
L
E"
is
set which
is n o t p(E',L)-
is o(L,E')-dense
in
is not p(E',L)-equi-
is w(E',L)-equicontinuous,
457
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
n = 1,2,..., THEOREM
9.
and, t h e r e f o r e ,
i s a d u a l metric space,
E
If
i s n o t a (DF)-space.
E'[b(E',L)]
i s a semi-
E'[X(E',E)]
S u s l i n space. PROOF.
Since
has a c o u n t a b l e f u n d a m e n t a l s y s t e m of bounded s e t s ,
E
i t follows t h a t
E'[B (E' , E ) ]
c o u n t a b l e bounded s e t and, t h e r e f o r e ,
A
A
i s metrizable.
in
1.9.
If
i s r e l a t i v e l y compact i n
E
every
i s equicontinuous i n
E'[p(E',E)]
3.
c l u s i o n f o l l o w s u s i n g Theorem COROLLARY
Moreover,
E'[X(E',E)].
E
The con-
q.e.d.
i s a Frechet
space,
E"[X (E" , E l
)]
i s a semi-
S u s l i n space. PROOF.
u s e Theorem
i s a (DF)-space,
E'[B(E',E)]
Since
[4], it suffices t o
9 i n o r d e r t o o b t a i n the c o n c l u s i o n .
THEOREM 1 0 .
Let
E
be a dual metric space.
m e t r i z a b l e s u b s e t of
E'[
1 (E'
,E)]
,
If
q.e.d. A
i s a closed
i s separable.
A
PROOF. C o n s i d e r i n g Theorem 9 and Lemma 1 i t f o l l o w s t h a t A [ h ( E ' , E ) ]
i s a K-Suslin
space and,
COROLLARY 1 . 1 0 .
A
therefore,
i s separable.
Let
E
be a Frechet space.
m e t r i z a b l e s u b s e t of
E?'[
X (E" ,E' )]
NOTE
3.
,
A
If
A
q.e.d.
i s a closed
i s separable.
L e t u s m e n t i o n t h a t t h e o r e m of J. Diedonne e s t a b l i s h i n g
t h a t e v e r y FrLchet-Monte1 s p a c e i s s e p a r a b l e i s a p a r t i c u l a r c a s e of our C o r o l l a r y 1 . 1 0 . The p r o o f
o f t h e n e x t theorem n e e d s t h e f o l l o w i n g r e s u l t ,
which we g i v e i n [ l l ] . t h e r e i s on
E
a ) Let
b e a l o c a l l y convex s p a c e .
a m e t r i z a b l e l o c a l l y convex t o p o l o g y
than t h e o r i g i n a l topology, E[U(E,E')]
E
i s compact i n
3
If
coarser
t h e n e v e r y c o u n t a b l y compact s u b s e t o f
E[u(E,E')]
.
458
M.
THEOFU3M 11.
Let
VALDIVIA
be a d u a l metric space.
E
E.
U-compact g e n e r a t e d s u b s p a c e of K-Suslin
Then
Let
F
b e a weakly
F'[X(F',F)]
is a
topological space.
PROOF.
S i n c e t h e completioii o f a d u a l m e t r i c s p a c e i s i t s e l f d u a l
metric,
l e t u s suppose t h a t
+;he c l o s u r e of p.
F
in
to
G,
let
Theorem
f
If
9 , t h e r e i s a mapping
Definition
are verified.
in
n = 1,2,,..
I) = f o v .
Let x
(v,)
Then
which l i e s i n
q(x)
(E '/G' ) [ k( E'/ G ', G ) ]
into the
U s i n g P r o p o s i t i o n 1, of
P,
Let u s t a k e a sequence
b e a sequence i n (v,)
Ry
E'/GL.
P
i s an a r b i t r a r y p o i n t
(E'/GL)[h(E'/GL,G)].
Let
$(x).
onto
E'
s u c h t h a t c o n d i t i o n s 1 and 2 of
i t i s enough t o show t h a t i f i s compact i n
E'
from a P o l i s h s p a c e
Cp
,E)]
E'[X(E'
151,
i s t h e o r t h o g o n a l s u b s p a c e of
Gi
b e t h e c a n o n i c a l mapping from
c l o s e d s u b s e t s of
be
G
a t o t a l s e q u e n c e of weakly compact
G
a b s o l u t e l y convex s e t s .
Let
I t f o l l o w s from a ,5eorem of I i r e i n ,
E.
325, t h a t t h e r e i s i n
i s a complete s p a c e .
E
h a s an a d h e r e n t p o i n t i n
and t h e r e f o r e
(u,)
which l i e s i n
$(x),
(u,)
f(vn) = u
such t h a t
q(x)
$(x)
n'
E'[X(E',E)]
has an a d h e r e n t p o i n t i n hence
is rela-
$(x)
t i v e l y c o u n t a b l y compact i n t h i s s p a c e and u s i n g r e s u l t a ) i t f o l lows e a s i l y t h a t F' with
X(E'/GL,G)-compact.
is
can be i d e n t i f i e d w i t h Since
),(F',F).
follows t h a t NOTE
$(x)
4.
I n [S]
H.
and t h e t o p o l o g y
E'/GL
h(E'/GL,F)
F'[X(F',F)]
On t h e o t h e r h a n d ,
i s coarser than
i s a K-Suslin
X(E'/G',F) L(E'/GL,G)
space.
P f i s t e r h a s proved t h a t i n a
it
q.e.d. (DF)-space e v e r y
p r e c o m p a c t s u b s e t i s s e p a r a b l e and m e t r i z a b l e i n t h e c a n o n i c a l u n i formity.
We c a n o b t a i n t h i s r e s u l t f o r d u a l m e t r i c s p a c e s u s i n g
o u r Theorem 11 i n t h e f o l l o w i n g way:
s e t of a d u a l m e t r i c s p a c e convex h u l l of
A
l i n e a r h u l l of
B.
E,
let
i n t h e completion Then
F
If B
.
E
A
i s a precompact sub-
be t h e closed a b s o l u t e l y of
E.
Let
F
be t h e
i s weakly compact g e n e r a t e d s p a c e i n -
459
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
I?, s o t h a t
s i d e a d u a l m e t r i c space
L
b e t h e tOp0106T on
space.
Let
on
Therefore,
B.
t h e compact s e t
F'[LI]
F'
i s a K-Suslin
of t h e u n i f o r m c o n v e r g e n c e
i s metrizable,
h e n c e s e p a r a b l e and s o
It fol1.ows t h a t
i s metrizable.
B
F'[X(F',F)]
is separable
A
and m e t r i z a b l e i n t h e c a n o n i c a l u n i f o r m i t y .
BIBLIOGRAPHY 1.
AMIR,
D.
and J . LINDESTRAUSS:
s e t s i n Banach s p a c e s , Ann. 2.
BESSAGA, C .
and S . ROLEWICZ:
'3,
C o l l o q . Math.
3.
DIEUDON&,
Acad.
4.
5. 6.
(1964).
S u r l e s th6orhmes d e S . Banach e t L.
MARTINEAU, A . :
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PFISTER, H . :
PIETSCH, A.:
VALDIVIA,
M.:
VALDIVIA, M . :
Moliner, s.n.
SPAIN
-
(1968).
Valencia
(1976).
Berlin-Heidelberg-
1972.
Sobre una c i e r t a c l a s e d e e s p a c i o s t o p o l 6 g i c o s .
(1977).
Some c r i t e r i a f o r weak c o m p a c t n e s s .
m, 165-169
F a c u l t a d de Matem6ticas Burjasot
Schwartz
30, 43-51
Arch. d . Math. 2 6 , 86-92
N u c l e a r L o c a l l y Convex S p a c e s .
angew. Math.
Dr.
S t u d i a Math.
B e n e r k i n g e n zum S a t s fiber d i e S e p a r a b i l i t a t
C o l l e c t a n e a Math. 2 8 , 9-20 11.
Berlh-Heildelberg-
1969.
122, 150-162
New York, S p r i n g e r :
10.
Summa B r a s i l .
(195'+).
d e r F$echet-Montel-R8ume.
9.
C.R.
(1954).
Some examples o n l i n e a r t o p o l o g i c a l s p a c e s .
KOMURA, Y . :
concernant
8.
298, 19'+-195
Springer:
Math. Ann.
7.
O n bounded s e t s i n F - s p a c e s .
T o p o l o g i c a l Vector Spaces I ,
New York,
35-46 ( 1 9 6 8 ) .
89-91 ( 1 9 6 2 ) .
S u r l e s e s p a c e s (F) e t (DF).
2, 57-122
KOTHE, G . :
( 2 ) 88,
S u r l e s e s p a c e s d e Monte1 s e p a r a b l e s .
J.:
Sci. Paris
GROTHENDIECK, A . : Math.
The s t r u c t u r e o f weakly compact of Math.
(1972).
J. r e i n e
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
Manuel V a l d i v i a
I n t h i s p a p e r we s t u d y some p r o p e r t i e s of a c l a s s of t o p o l o g i c a l s p a c e s i n c l u d i n g t h e K-Suslin
s p a c e s and h e n c e f o r t h we o b t a i n
some new r e s u l t s i n t h e t o p o l o g i c a l v e c t o r s p a c e s t h e o r y .
A i s a s t a r s h a p e d m e t r i z a b l e subse-t of a
l a r l y , we p r o v e t h a t i f Hausdorff
E,
topological v e c t o r space
E
bounded s u b s e t of
Particu-
intersects
A
s u c h t h a t e v e r y c l o s e d and
i n a compact s e t , t h e n
A
is
separable.
We u s e t h e r e H a u s d o r f f t o p o l o g i c a l s p a c e s . v e c t o r s p a c e s u s e d h e r e a r e d e f i n e d on t h e f i e l d complex numbers. B(E,F)
and
respectively
3
If
(E,F)
If
W(E,F)
K
of t h e r e a l or
i s a d u a l p a i r we d e n o t e by
t h e weak,
.
The t o p o l o g i c a l
u(E,F),
s t r o n g and Mackey t o p o l o g i e s on
i s t h e t o p o l o g y of a t o p o l o g i c a l s p a c e
ACT]
T
and
E,
a s u b s e t of
T,
we d e n o t e by
t o p o l o g y by
3.
The t o p o l o g i c a l d u a l of a l o c a l l y convex s p a c e
is
E‘
.
X(E’,E)
E”
the s e t
i s t h e t o p o l o g i c a l d u a l of
t h e t o p o l o g y on
compact s u b s e t s of
E.
E’ On
with the
A
E’[B (E‘ , E ) ]
E
induced E
We d e n o t e by
of t h e u n i f o r m c o n v e r g e n c e on t h e p r e -
E”
,
X(E” ,E’
)
i s t h e t o p o l o & T of t h e
u n i f o r m c o n v e r g e n c e on t h e p r e c o m p a c t s u b s e t s of u s u a l , we i d e n t i f y
.
is
A
w i t h a s u b s p a c e of
EN
E‘[B (E’ , E ) ]
.
As
by t h e c a n o n i c a l i n -
j e c t i on. By a “ w e a k l y u-compact
g e n e r a t e d l o c a l l y convex s p a c e ”
G
446
M.
VALDIVIA
we mean a l o c a l l y convex s p a c e
which h a s a s e q u e n c e of weakly
G
compact s u b s e t s whose u n i o n i s t o t a l i n of bounded
of
B
s o that i f
I.
H
z
H
E B,
i s contained i n
{txo
+
(1-t)z
: 0 i t
A t o p o l o g i c a l space
a P o l i s h space s e t s of
2.
E
space
{x,]
and a mapping
x
c
I.
in
B
l] C B .
P of
cp
and a mapping
u
{cp(x) : x
c
cp
from
P
there exists
i n t o t h e c l o s e d sub-
in
P
PROPOSITION 1.
so that
t h e r e i s a P o l i s h space E,
then
from
V
P
(zn)
h a s an adherent p o i n t
i f there is a Polish
i n t o t h e compact s u b s e t s of F ,
and g i v e n a n a r b i t r a r y p o i n t of
z
t h e r e i s a neighbourhood
cp(z),
[7].
cp(U) C V ,
P
converging towards an element
i s K-Suslin
F
A topological
compact s u b s e t s of
P
cp(x).
P] = F
and a n e i g h b o u r h o o d z
i s semi-Suslin i f
n = 1,2,...,
which i s c o n t a i n e d i n
P
[lo]:
E
i s a seqaence i n
zn E cp(xn),
such t h a t
U
i
such t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :
If
and E
P
A topological space
in
I]
SEMI-SUSLIN SPACES
DEFINITION.
in
2
c
is f u n d a m e n t a l
i s starshaped i f t h e r e i s a vector then
: i
{Ai
for some
A.
The f o l l o w i n g d e f i n i t i o n was g i v e n i n
x
H
s e t s i n t h e topological vector space
i f e a c h bounded s e t i n A subset
A family
G.
space
E
i s K-Suslin
and a mapping
cp
from
i f and o n l y i f P
into the
s u c h t h a t t h e two f o l l o w i n g c o n d i t i o n s a r e
satisfied:
2.
x
and
If zn
(x,)
i s a sequence i n
E cp(x,),
n = 1,2,...,
P
c o n v e r g i n g t o w a r d s an e l e m e n t
then
(zn)
h a s an a d h e r e n t p o i n t
447
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
in
which i s c o n t a i n e d i n
E
PROOF.
Let u s suppose t h a t E
p r o p o s i t i o n and x
E
P,
U
of
(Un)
L e t u s t a k e now a p o i n t
z n E cp(xn),
g
zn
t i o n says t h a t cp(x).
E
v e r i f i e s t h e two c o n d i t i o n s of t h e
i s n o t a K-Suslin
a neighbourhood
neighbourhoods
to
cp(x).
xn
in
E
(zn)
P
Un
n = 1,2,
U,
~ ( x ) and a f u n d a m e n t a l s y s t e m of
of x
Then t h e r e i s a p o i n t
space.
z
n=1,2,... and a
The c o n d i t i o n 2 of t h e p r o p o s i -
z
h a s an a d h e r e n t p o i n t
On t h e o t h e r h a n d ,
U,
cp(xn) q! U,
such t h a t
...
~ ( u , )$
such t h a t
6
in
~ ( x )because
E zn
which b e l o n g s
E
U,
n=1,2
,...
T h e r e f o r e we a r r i v e t o a c o n t r a d i c t i o n . E
C o n v e r s e l y , l e t us suppose t h a t a mapping from a P o l i s h s p a c e
u
that
and a n e i g h b o u r h o o d U
of
= E
(cp(x) : x f P]
z
in
P
of
V
such t h a t
i s K-Suslin.
i n t h e compact s u b s e t s o f
P
~ ( z )i n
E,
cp(U) c V .
x
and a s e q u e n c e
which h a s n o t a d h e r e n t p o i n t i n
compact, i f
M
in
cp(x).
,...,zn ,...3 ,
M
t e and t h e r e f o r e t h e r e e x i s t s a p o s i t i v e i n t e g e r
... ] n cp(x)
( ~ ~ ~ , z ~ ~ + ~ E, v.i d. e. n]t l.y , bourhood
X
of
x
positive integer znl
E
B fl cp(x
such t h a t n1
n
B
Let
~ ( x = ) $,
B
) c B fl q ( X ) = $ ,
in
Since
n
no
cp(x)
On t h e o t h e r h a n d ,
which a r e n o t K-Suslin
[lo].
P
cp(x) i s is fini-
such t h a t
hence t h e r e i s a neighWe c a n c h o o s e now a
so that
x
Therefore
P r o p o s i t i o n 1 allow u s t o o b t a i n t h a t e v e r y K-Suslin i s semi-Suslin.
does
zn 6 ~ ( x , ) ,
E,
E X. "1 hence a c o n t r a d i c t i o n .
n
P
b e t h e c l o s u r e of
cp(X) r7 B = $ .
l a r g e r than
"1
= Q.
(x,)
(zn)
n = 1,2,...
no+P'
in
cp
Let u s suppose t h a t
which converges towards
rzno,zno+l,...,z
such
t h e r e i s a neighbourhood
Then t h e r e i s a s e q u e n c e
{z1,z2
be
E
z
and g i v e n a n a r b i t r a r y p o i n t
n o t v e r i f y c o n d i t i o n 2.
i s the s e t
cp
Let
q.e.d. space
t h e r e a r e semi-Suslin spaces
448
M.
LEMMA 1.
Let
F
VALDIVIA
be a s e m i - S u s l i n t o p o l o g i c a l s p a c e .
F
m e t r i z a b l e c l o s e d s u b s p a c e of
then
E
If
is a
E
i s K-Suslin.
PROOF.
S i n c e e v e r y c l o s e d s u b s p a c e of a s e m i - S u s l i n s p a c e i s semi-
Suslin,
[lo],
w e t a k e a mapping
L e t us t a k e i n
compact.
n = 1,2,.
..
cp(x)
Let
(U,)
t h e sequence
P,
i n the
(zn)
~ ( x )i s
then
(xn)
xn = x ,
siich t h a t
i s a n a r b i t r a r y sequence i n
cp (x)
,
zn F cp(xn),
has an a d h e r e n t p o i n t i n
i s c o u n t a b l y compact.
results that
Let
P
and t h e r e f o r e
rp(x)
LEMMA 2 .
(zn)
If
n = 1,2,..., Hence
i s an a r b i t r a r y point i n
x
P
T a k i n g a c c o u n t of P r o p o s i t i o n 1, i t s u f f i c e s t o
tion are verified. show t h a t i f
from a P o l i s h space
s o t h a t t h e c o n d i t i o n s 1 and 2 of t h e d e f i n i -
E
c l o s e d s u b s e t s of
rp
Since
i s metrizable i t
E
i s compact.
q.e.d.
b e a s u b s e t of a t o p o l o g i c a l v e c t o r s p a c e
A
rp(x).
E
b e a s e q u e n c e o f c l o s e d c i r c l e d s u b s e t s of
E[Z].
which v e -
r i f i e s t h e following conditions:
1.
If
z
an a r b i t r a r y p o i n t
of
a r e given t h e r e i s a positive i n t e g e r 2.
If
(mp)
and a p o s i t i v e i n t e g e r
A
n
P
such t h a t
z
E
p
n U P P'
i s a n a r b i t r a r y s e q u e n c e of p o s i t i v e i n t e g e r s t h e
set
[n i s non-void
Then PROOF.
Let
Empup
= i,2, ...I]
n
A
and c o u n t a b l y compact. A[3] N
i s a semi-Suslin space.
b e t h e s e t of p o s i t i v e i n t e g e r numbers w i t h t h e
d i s c r e t e topology.
By
NN
of c o u n t a b l e many c o p i e s of
u s c o n s i d e r t h e mapping A[J]
: P
rp
we r e p r e s e n t t h e t o p o l o g i c a l p r o d u c t N.
Then
from
NN
NN
i s a P o l i s h space.
i n t o t h e c l o s e d s u b s e t s of
such t h a t i f
x = (x1,x2
Let
,...,xP' ...) E
NN
44 9
ON SEMI-SUSLIN S P A C E S AND DUAL METRIC SPACES
then
CP(~)rn Expup :
...I] n
P = i,2,
A.
Using condition 1 of this lemma it follows that A = [pp(x)
(x'"))
converges towards
If
cp(x(")). teger
n(p)
p
2
x
in
NN
NN.
such that the sequence zn
Let us take a point
such that
x(~) = x
P (x("))
for every P' towards x in
n 2 n(~),
NN.
because
Then, if
n(p), zn E cp(x ( n ) ) c xPuP
{.An)
...I
: n = 1,2,
then
zn
E
[n CYpup
: P = i,2,
and since this set is countably compact, point
in
zo
A[3]
by (1).
has an adherent
...!I
: P = i,2,
n
A = rp(x)
q.e.d.
THEOREM 1.
Let
be a starshaped metrizable subset of a topolo-
A
E[3].
gical vector space intersects
E
(zn)
A
Therefore conditions 1 and 2 of the definition are sa-
tisfied.
PROOF.
...11 n
which belongs to
3? Expup
E[3]
(1)
is the maximum of the finite set of natural numbers
If yp
of
in
is a positive integer there exists a positive in-
of the convergence of n
,...,xP( " ) ,...) E
= (xin),x$n)
x(")
Let
N
: x E N ).
A
If every closed and bounded subset of
in a compact set,
A[3]
is a K-Suslin space.
Obviously, it sufficos - t o prove the theorem when the origin belongs to
point of
A
A
and every segment which joints an arbitrary
with the origin lies in
there is a sequence
(U,)
A.
Since
A
is metrizable
of closed circled neighbourhoods of the
M. VALDIVIA
450
origin i n
such t h a t
E[3]
(un n is
R
f u n d a m e n t a l s y s t e m of
A : n=1,2,
...I
n e i g h b o u r h o o d s of t h e o r i g i n i n
O b v i o u s l y c o n d i t i o n 1 o f Lemma 2 i s v e r i f i e d on
(Un).
If
A[3]. (mp)
i s a n a r b i t r a r y s e q u e n c e of p o s i t i v e i n t e g e r s t h e s e t
i s non-void,
because t h e o r i g i n has i n i t .
To p r o v e t h i s l e t
t h i s s e t i s compact.
bourhood of t h e o r i g i n i n
q
Let u s s e e now t h a t
V
b e an a r b i t r a r y neigh-
Then t h e r e i s a p o s i t i v e i n t e g e r
E[3].
such t h a t
u 9 n ~ c v An. If
z
E
n
(mqUq)
A,
n
E Uq
then
A
and i t f o l l o w s t h a t
¶
[n
Empup
: p=1,2,
and t h e r e f o r e ,
...I] n
A C
( m9 u9 ) n
A C m ( U nn) q 9
t h e s e t ( 2 ) i s bounded i n
E[3].
i t i s evident t h a t t h i s s e t i s closed i n i s compact. verified.
I t allows u s t o conclude t h a t
PROOF.
v
9
O n t h e o t h e r hand,
and, t h e r e f o r e ,
it
intersects
A[%]
i s a semi-Suslin
i s K-Suslin.
A[3]
b e a s t a r s h a p e d and m e t r i z a b l e s u b s e t o f a t o -
A
Let
pological vector space E[J]
9
We h a v e t h u s p r o v e d t h a t c o n d i t i o n 2 of Lemma 2 i s
s p a c e , and by Lemma 1, THEOREM 2 .
A[3]
c rn ( V n A ) c m
E[3].
If
e a c h bounded and c l o s e d s u b s e t of
i n a compact s e t ,
A
By Theorem 1,
A[3]
i s K-Suslin
A[3]
i s separabla.
and t h e r e f o r e L i n d e l 8 f , [ 7 ] .
Since every metrizable Lindelbf space i s separable, complet e , COROLLABY 1 . 2 .
t h e proof
is
q.e.d. Let
g i c a l v e c t o r space
A
E[3].
b e a m e t r i z a b l e convex s u b s e t o f a t o p o l o I f e v e r y bounded and c l o s e d s u b s e t of
451
ON SEMI-SUSLIN SPACES AND DUAL METRIb SPACES
E[3]
intersects
NOTE 1.
A
in a compact set,
A. Grothendieck asks in
A[s]
is separable.
[4] if every FrBchet-Monte1 space
J. Dieudonne gave an affirmative answer to this
is separable.
.
question in [3]
C. Bessaga and S. Rolewicz proved in [ 2 ]
that
every metrizable Monte1 topological vector space is separable. This result can be obtained from our Corollary 1.2 taking THEOREM
3
3.
Let
A = E.
be a metrizable topological vector space.
E
be a topological vector t o p o l o g on
E
Let
coarser than the ori-
ginal topology such that the following conditions are satisfied: 1.
There is a fundamental system of neighbourhoods of the E
origin of 2.
which are closed in
E[3].
Every bounded subset of E is relatively countably compact in
EC3l. E[3]
Then PROOF.
is a semi-Suslin topological space.
(Un) be a fundamental sequence of circled neighbour-
Let
hoods of the origin in A = E
E,
which are closed in
that lemma is satisfied.
On the other hand, let
quence of positive integer numbers.
n compact.
E[3]
(mp)
be a se-
The set
Empup : p=i,2, ...I
and closed in
E[3]
and, therefore, 3-countably
Consequently, condition 2 is satisfied.
It follows that
is a semi-Suslin space.
THEOREM 4. 3
E
Let us take
I t follows straightforward that condition 1 of
in Lemma 2 .
is bounded in
E[3].
Let
E
be a metrizable topological vector space.
be a topological vector topology on
E
Let
coarser than the original
topology such that the following conditions are satisfied:
1.
There exists a fundamental system of neighbourhoods of the
origin in
E
which are closed in
E[3].
452
M.
E v e r y bounded s e t i n
2.
Then
E[3]
i s a K-Suslin
i s r e l a t i v e l y compact i n
E[3].
topological space.
of Lemma 2 i s s u c h t h a t
b u i l t i n t h e proof
x E N
f o r every
x = (x1,x2,
E
EC31.
9.e.d.
THEOREM
5.
If
.
rp(x)
i s compact i n
It follows s t r a i g h t f o r w a r d , s i n c e i f
the s e t
and c l o s e d i n
E
and t h e r e f o r e compact i n
E[3]
i s a Fr6chet space,
E'[b(E'
i f and o n l y i f PROOF.
N
...,xn, ...)
i s bounded i n
A
E
By P r o p o s i t i o n 1, i t s u f f i c e s t o p r o v e t h a t t h e mapping
PROOF.
E[3]
VALDIVIA
,EN)]
E"[X(E",E')]
i s K-Suslin
i s barrelled.
L e t us s u p p o s e f i r s t t h a t
i s K-Suslin.
E"[X(E" , E ' ) ]
Let
b e a n a b s o l u t e l y corivex c l o s e d and bounded s u b s e t o f E " [ u ( E " , E ' ) ] .
G r o t h e n d i e c k proved t h a t e v e r y c o u n t a b l y s u b s e t of tinuous i n E"[u(E"
Since
,E')].
E"[a (E" ,E' )]
[4],
[ 71
E'[w (E' ,E"
hence
.
)]
and t h e r e f o r e Hence
E"[p(E" , E ' ) ] ,
S i n c e each sequence i n
i t follows t h a t
).(E",E')
A
i s a Lindel8f
i s u(E" ,E' )-compact and c o n s e -
A
E'[u(E'
,E")]
is barrelled.
A
and
u(E",E') A
If
E ' [ p (E' , E ) ]
E
is
[S]
,
c o i n c i d e s on t h i s sequence,
i s r e l a t i v e l y compact i n
By u s i n g Theorem 4 w e o b t a i n t h e c o n c l u s i o n . Let
A
i s r e l a t i v e l y u(EN, E ' ) - c o m p a c t .
i s equicontinuous i n
hence i t i s e a s y t o o b t a i n t h a t
COROLLARY 1.5.
A[a (E" , E ) ]
of t h e o r i g i n which a r e h ( E " , E ' ) - c l o s e d .
a bounded s e t i n
,E')].
it follows t h a t
i s a F r 6 c h e t s p a c e which h a s a f u n d a m e n t a l s y s t e m of
neighbourhoods
E"[X(E"
i s c o u n t a b l y compact i n
i s barrelled.
L e t u s s u p p o s e now t h a t
E"[p(E",E')]
A
i s K-Suslin
E"[h(E" , E ' ) ]
i s K-Suslin
t o p o l o g i c a l space quently
,
E'[@(E' ,E)]
i s equicon-
A
be a Fr6chet space.
If
E
q.e.d.
i s distinguished
453
ON SEMI-SUSLIN SPACES AND DUAL &QCTRICSPACES
then
I n [4]
NOTE 2.
G
(EN ,E' )]
E"[
i s K-Suslin. G r o t h e n d i e c k g i v e s a n e x a m p l e s of a F r 6 c h e t s p a c e
A.
G'[p (G' ,G" )]
such t h a t
i s n o t K-Suslin
i s not barrelled.
H
example o f a F r 6 c h e t s p a c e $(HI ,H)
and
f
,*). H
w(H'
g ' [ X ( ~, H ' ')]
THEOREM 6.
E
Let
PROOF. that
If
such t h a t
Komura g i v e s i n
E"[X(E",E')]
i s K-Suslin
i s Lindelbf. A
and t h e r e f o r e
E"[X(E"
Since
,E')]
is
E"[
i s an
A
(E" ,E' ) ]
(E" ,E' ) - c o u n t a b l y
,
compact
Hence E'[p (E' ,E" )]
q.e.d.
DUAL METRIC SPACES A l i n e a r topological
if
A
is
E"[X(E",E')]
i s L i n d e l b f and
i s X (EN ,E' ) - c o m p a c t .
is barrellled.
11.
5
i s b a r r e l l e d i t f o l l o w s f r o m Theorem
E'[p(E',E")]
[ 41 i t f o l l o w s t h a t
i s Lindelbf
is barrelled.
E'[p(E',E")]
Conversely, i f
)]
an
i s barrelled
H'[M(H',H")]
a b s o l u t e l y convex c l o s e d and boJnded s u b s e t o f
A[A (E" ,E'
[6]
i s K-Suslin.
be a Fr6chet space.
E"[X(E",E')]
Lindelbf.
Y.
i s a n example of a n o n - d i s t i n g u i s h e d
space such t h a t
i f and o n l y i f
5.
b e c a u s e o f Theorem
T h e r e f o r e G"[X (G" ,G')
l o c a l l y convex s p a c e
E
i s dual metric
i t h a s a c o u n t a b l e f u n d a m e n t a l s y s t e m of bounded s e t s and i n E'
each
3 (E' ,E)-bounded s e q u e n c e i s e q u i c o n t i n u o u s [ 91 , p . 11. A l i n e a r t o p o l o g i c a l l o c a l l y convex s p a c e
E
i s (DF)
h a s a c o u n t a b l e f u n d a m e n t a l s y s t e m of bounded s e t s a n d i n $(E',E)-bounded
E'
i f it each
s e t which i s c o u n t a b l e u n i o n of e q u i c o n t i n u o u s s e t s
i s i t s e l f equicontinuous [4]. Obviously,
e v e r y (DF)-space i s d u a l m e t r i c .
The f o l l o w i n g
two t h e o r e m s g i v e some c l a s s e s o f d u a l m e t r i c s p a c e s which a r e n o t (DF) *
454
M.
THEOREM
7.
pology
3
If
compatible with the dual p a i r
E',
(E,E')
such t h a t
i s a d u a l m e t r i c s p a c e which i s n o t (DF).
E'[3]
Let
PROOF.
Since
i c
I]
b e a maximal o r t h o n o r m a l s y s t e m i n
i s not separable t h e r e i s a p a r t i t i o n of
E
...,I n ,...
11,12,
n = 1,2,... : i
:
(xi
many s u b s e t s
(xi
i s a non-separable H i l b e r t space t h e r e i s a t o -
E
on
VALDIVIA
Let
6 In]
In
i n countable
i s not countable,
b e t h e c l o s e d a b s o l u t e l y convex h u l l of
An
in
such t h a t
I
E.
E.
We d e n o t e by
@
a l l t h e s u b s e t s of
E
of
t h e form m
with
a b s o l u t e l y convex bounded and s e p a r a b l e and
A
n i t e l y many n o n - z e r o p o l o g y on
E'[3]
3
If
3
i s t h e to-
i s compatible w i t h t h e d u a l p a i r
i s a dual metric space.
(DF)-space because
3
{A,
: n=1,2,
Moreover,
...I
i t i s not i t s e l f equicontinuous.
(E,E')
i s not a
E'[3]
E'[3]
7 a r e not
Theorem 8 g i v e s Mackey d u a l m e t r i c s p a c e s which a r e
(DF).
F o r t h e n e x t theorem l e t u s t a k e a Banach s p a c e t h e r e i s a n o n - s e p a r a b l e a b s o l u t e l y convex and we-kly set
and
q.e.d.
The d u a l m e t r i c s p a c e s o b t a i n e d u s i n g Theorem Mackey s p a c e s .
8,
i s a s t r o n g l y bounded s e t
w h i c h i s a c o u n t a b l e u n i o n of e q u i c o n t i n u o u s s e t s i n
not
a fi-
o f t h e u n i f o r m c o n v e r g e n c e on t h e e l e m e n t s of
E'
i t i s evident t h a t and
s e q u e n c e of r e a l numbers.
(X,)
X
in
F.
Let
E
t i n u o u s f u n c t i o n s from
b e t h e Banach s p a c e X[o(F,F')]
into
F
compact s u b -
C(X[O(F,F')])
K,
so that
of con-
w i t h t h e u n i f o r m con-
vergence topology. THEOREM 8. such t h a t
In
E"
t h e r e i s a v e c t o r subspace
E'[u(E',L)]
L
containing
i s a non-(DF) d u a l m e t r i c s p a c e .
E
455
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
PROOF.
Let
M
b e t h e l i n e a r h u l l of
is
M
t h a t t h e o r i g i n of
in
Gb
X
in
X[a(F,F')].
.
m
n
Let u s suppose Let
b e a se-
(Un)
of t h e o r i g i n i n
q u e n c e of a b s o l u t e l y convex n e i g h b o u r h o o d s X[u(F,F' )]
F.
Un = { O } For each p o s i t i v e i n t e g e r p n= 1 { p Un : n = 1 , 2 , ...I i n t h e f a m i l y of n e i g h b o u r h o o d s of t h e o r i g i n
such t h a t
( p X)[u(F,F')]
h a s t h e same p r o p e r t y .
Given t h e p a i r
positive integers there exists a f i n i t e set
A
in
P"
(p,n) F'
of
such t h a t
Ao n p X c p U n pn
being
t h e p o l a r s e t of
Ao P"
If
P
A
in
Pn
i s t h e l i n e a r h u l l of
(M,P)
i s a d u a l p a i r and
Since
P
u(M,P)
F.
: p , 1 , 2 ,...)
(Apn
coincides with
u(F',F)
h a s a c o u n t a h l e Hamel b a s i s i t f o l l o w s t h a t
i s m e t r i z a b l e and, t h e r e f o r e , s e p a r a b l e .
in
F', in
X.
X[u(F,F')]
But t h i s a c o n t r a d i c t i o n
w i t h the h y p o t h e s i s .
It permits t o a s s e r t t h a t t h e r e i s a point which i s n o t
X[u(F,F')]
s y s t e m of n e i g h b o u r h o o d s of f u n c t i o n from fi(xo) we t a k e
= 1,
X[o(F,F')]
= 0,
fi(x)
a bounded n e t i n
E.
x E X-
(g,ux)
= g(x),
(f,ux)
= 0, Let
H E 51,
let
51 H*
If
in
be a f u n d a m e n t a l fi
[O,l]
il,iz E I (fi
be a continuous such t h a t and vil : i E I, S
t h e r e i s a element
f
in
c v
iz is
)
E"
t o this net.
let
f o r every
I}
and l e t
X
vi.
Consequently,
x 6 X,
x E X-
in
E
i s a d i r e c t s e t and
w h i c h i s a(E" ,E' ) - a d h e r e n t For each
xo
: i
(Vi
i n t o the i n t e r v a l
(I, s)
iz i il.
Let
Gb.
x
ux
b e t h e e l e m e n t of
g E C(X[u(F,F')]).
such t h a t
E'
Obviously,
(f,uXO)=l,
(xo3.
b e the f a m i l y of a l l t h e c o u n t a b l e s e t s i n b e t h e c l o s u r e of
H
in
E"[u(E" ,E')]
.
E. Let
If
456
M. VALDIVIA
L
is a subspace of
E"
containing
Then there is a sequence
,E' ) ]
E"[u(E"
point in
wn The set r) {W,
(g,)
.
which has
E
f
as adherent
Let
x
= cx E
: n=1,2,.
..]
{x,]
is different from
and therefore
such that 1 n
<
Ign(xl)-gn(xo)
1 r ~ .
<
: ign(x)-gn(xo)i
x1 f xo
x1 E X,
there is a point
in
Let us suppose that f E L.
E.
11
9
= 1,2,...
It follows that
(f,ux ) 1
-
and this gives a contradiction.
A
set of H*
3
0
i
= 0
Therefore,
E'[t.i(E',L)]
Let us see now that Let
(f,ux
f
{
L.
is a dual metric space.
be a separable absolutely convex, closed and bounded sub-
aU(L,E')].
A,
hence
A
is u(L,E')-compact
equicontinuous and so,
E'[k(E,L)]
and therefore
such that is p(E',L)-
A
is a dual metric space.
In the sequel we shall prove that (DF)-space.
H € 51
Then there is an element
E'[@(E',L)]
is not a
B y using a result of Amir and Lindestrauss [l] there
D
of
is dense in
B.
is an absolutely convex weakly compact and total subset Let
B
be the closed unit ball of
E.
E.
Let
Bn = B n n D. Bn
is weakly compact and
L
Since
is different of
not contained in
L,
B*
n
L
B*
n
L
u [Bn : n=1,2, ...] that u [Bn : n=1,2, . . . I
On the other hand,
Bn
B*
of
B
in
is a 8 L,E')-bounded
and, consequently,
Since
it follows
continuous.
{Bn : n=1,2,.. ]
E N , the closure
hence
is not g(L,E')-cornpact equicontinuous.
u
BY
n
L
E"
is
set which
is n o t p(E',L)-
is o(L,E')-dense
in
is not p(E',L)-equi-
is w(E',L)-equicontinuous,
457
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
n = 1,2,..., THEOREM
9.
and, t h e r e f o r e ,
i s a d u a l metric space,
E
If
i s n o t a (DF)-space.
E'[b(E',L)]
i s a semi-
E'[X(E',E)]
S u s l i n space. PROOF.
Since
has a c o u n t a b l e f u n d a m e n t a l s y s t e m of bounded s e t s ,
E
i t follows t h a t
E'[B (E' , E ) ]
c o u n t a b l e bounded s e t and, t h e r e f o r e ,
A
A
i s metrizable.
in
1.9.
If
i s r e l a t i v e l y compact i n
E
every
i s equicontinuous i n
E'[p(E',E)]
3.
c l u s i o n f o l l o w s u s i n g Theorem COROLLARY
Moreover,
E'[X(E',E)].
E
The con-
q.e.d.
i s a Frechet
space,
E"[X (E" , E l
)]
i s a semi-
S u s l i n space. PROOF.
u s e Theorem
i s a (DF)-space,
E'[B(E',E)]
Since
[4], it suffices t o
9 i n o r d e r t o o b t a i n the c o n c l u s i o n .
THEOREM 1 0 .
Let
E
be a dual metric space.
m e t r i z a b l e s u b s e t of
E'[
1 (E'
,E)]
,
If
q.e.d. A
i s a closed
i s separable.
A
PROOF. C o n s i d e r i n g Theorem 9 and Lemma 1 i t f o l l o w s t h a t A [ h ( E ' , E ) ]
i s a K-Suslin
space and,
COROLLARY 1 . 1 0 .
A
therefore,
i s separable.
Let
E
be a Frechet space.
m e t r i z a b l e s u b s e t of
E?'[
X (E" ,E' )]
NOTE
3.
,
A
If
A
q.e.d.
i s a closed
i s separable.
L e t u s m e n t i o n t h a t t h e o r e m of J. Diedonne e s t a b l i s h i n g
t h a t e v e r y FrLchet-Monte1 s p a c e i s s e p a r a b l e i s a p a r t i c u l a r c a s e of our C o r o l l a r y 1 . 1 0 . The p r o o f
o f t h e n e x t theorem n e e d s t h e f o l l o w i n g r e s u l t ,
which we g i v e i n [ l l ] . t h e r e i s on
E
a ) Let
b e a l o c a l l y convex s p a c e .
a m e t r i z a b l e l o c a l l y convex t o p o l o g y
than t h e o r i g i n a l topology, E[U(E,E')]
E
i s compact i n
3
If
coarser
t h e n e v e r y c o u n t a b l y compact s u b s e t o f
E[u(E,E')]
.
458
M.
THEOFU3M 11.
Let
VALDIVIA
be a d u a l metric space.
E
E.
U-compact g e n e r a t e d s u b s p a c e of K-Suslin
Then
Let
F
b e a weakly
F'[X(F',F)]
is a
topological space.
PROOF.
S i n c e t h e completioii o f a d u a l m e t r i c s p a c e i s i t s e l f d u a l
metric,
l e t u s suppose t h a t
+;he c l o s u r e of p.
F
in
to
G,
let
Theorem
f
If
9 , t h e r e i s a mapping
Definition
are verified.
in
n = 1,2,,..
I) = f o v .
Let x
(v,)
Then
which l i e s i n
q(x)
(E '/G' ) [ k( E'/ G ', G ) ]
into the
U s i n g P r o p o s i t i o n 1, of
P,
Let u s t a k e a sequence
b e a sequence i n (v,)
Ry
E'/GL.
P
i s an a r b i t r a r y p o i n t
(E'/GL)[h(E'/GL,G)].
Let
$(x).
onto
E'
s u c h t h a t c o n d i t i o n s 1 and 2 of
i t i s enough t o show t h a t i f i s compact i n
E'
from a P o l i s h s p a c e
Cp
,E)]
E'[X(E'
151,
i s t h e o r t h o g o n a l s u b s p a c e of
Gi
b e t h e c a n o n i c a l mapping from
c l o s e d s u b s e t s of
be
G
a t o t a l s e q u e n c e of weakly compact
G
a b s o l u t e l y convex s e t s .
Let
I t f o l l o w s from a ,5eorem of I i r e i n ,
E.
325, t h a t t h e r e i s i n
i s a complete s p a c e .
E
h a s an a d h e r e n t p o i n t i n
and t h e r e f o r e
(u,)
which l i e s i n
$(x),
(u,)
f(vn) = u
such t h a t
q(x)
$(x)
n'
E'[X(E',E)]
has an a d h e r e n t p o i n t i n hence
is rela-
$(x)
t i v e l y c o u n t a b l y compact i n t h i s s p a c e and u s i n g r e s u l t a ) i t f o l lows e a s i l y t h a t F' with
X(E'/GL,G)-compact.
is
can be i d e n t i f i e d w i t h Since
),(F',F).
follows t h a t NOTE
$(x)
4.
I n [S]
H.
and t h e t o p o l o g y
E'/GL
h(E'/GL,F)
F'[X(F',F)]
On t h e o t h e r h a n d ,
i s coarser than
i s a K-Suslin
X(E'/G',F) L(E'/GL,G)
space.
P f i s t e r h a s proved t h a t i n a
it
q.e.d. (DF)-space e v e r y
p r e c o m p a c t s u b s e t i s s e p a r a b l e and m e t r i z a b l e i n t h e c a n o n i c a l u n i formity.
We c a n o b t a i n t h i s r e s u l t f o r d u a l m e t r i c s p a c e s u s i n g
o u r Theorem 11 i n t h e f o l l o w i n g way:
s e t of a d u a l m e t r i c s p a c e convex h u l l of
A
l i n e a r h u l l of
B.
E,
let
i n t h e completion Then
F
If B
.
E
A
i s a precompact sub-
be t h e closed a b s o l u t e l y of
E.
Let
F
be t h e
i s weakly compact g e n e r a t e d s p a c e i n -
459
ON SEMI-SUSLIN SPACES AND DUAL METRIC SPACES
I?, s o t h a t
s i d e a d u a l m e t r i c space
L
b e t h e tOp0106T on
space.
Let
on
Therefore,
B.
t h e compact s e t
F'[LI]
F'
i s a K-Suslin
of t h e u n i f o r m c o n v e r g e n c e
i s metrizable,
h e n c e s e p a r a b l e and s o
It fol1.ows t h a t
i s metrizable.
B
F'[X(F',F)]
is separable
A
and m e t r i z a b l e i n t h e c a n o n i c a l u n i f o r m i t y .
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